This now seems more analogous to the situation that sometimes happens in rural Ireland. For example, in the area of West Cork that I often visit, an enormous proportion of the population is called O'Sullivan. (I recently saw the Facebook page of the Adrigole U15 Gaelic Football team who were playing against my nephews' team, and 3 of the 4 coaches, and 14 of the 15 players, were O'Sullivan.)
So an unofficial surname is added, which is not necessarily part of their legal name, and is only used locally. Thus there would be a Paddy Barry O'Sullivan, Paddy Jer O'Sullivan, and Paddy Joe O'Sullivan. Barry, Jer and Joe would not be on legal documents, and will have served as an unofficial patronym for many generations. These three Paddys might each have a brother called Michael Barry, Michael Jer and Michael Joe respectively, or even sisters called Mary Barry, Mary Jer and Mary Joe. In the village, it is not helpful to talk about whether you have seen Paddy O'Sullivan recently, but you can ask after Paddy Jer (and never bother to add the O'Sullivan bit), but the same guy will have a driving licence or passport that says (eg) Patrick Anthony O'Sullivan (his middle name, as per Christopher Clive Froome, is never used in conversation)
Is the Danish situation therefore similar, except that the middle name is included in documents and not limited to local use?